6.1: Compound Interest and Fundamentals
Formula & Symbol Hub
For this section you will need the following:
Symbols Used
- [latex]\text{C/Y}=[/latex] Compounds per year
- [latex]i=[/latex] Periodic interest rate
- [latex]\text{I/Y}=[/latex] Nominal interest rate per year
Formulas Used
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Formula 6.1 – Periodic Interest Rate
[latex]\begin{align*}i=\frac{\text{I/Y}}{\text{C/Y}}\end{align*}[/latex]
Compound Interest and Fundamentals
Compound interest is used for most transactions lasting one year or more. In simple interest, interest is converted to principal at the end of the transaction. Therefore, all interest is based solely on the original principal amount of the transaction. Compound interest, by contrast, involves interest being periodically converted to principal throughout a transaction, with the result that the interest itself also accumulates interest.
Calculating the Periodic Interest Rate
The first step in learning about investing or borrowing under compound interest is to understand the interest rate used in converting interest to principal. You commonly need to convert the posted interest rate to find the exact rate of interest earned or charged in any given time period.
[latex]\boxed{6.1}[/latex] Periodic Interest Rate
[latex]\color{red}{i\;}\color{black}{\text{is Periodic Interest Rate:}}[/latex] The percentage of interest earned or charged at the end of each compounding period is called the periodic interest rate. You calculate it by taking the nominal interest rate and dividing by the compounding frequency. For example, [latex]12\%[/latex] compounded quarterly has a periodic interest rate of [latex]12\%\div 4=3\%[/latex]. This means that at the end of every three months, you calculate [latex]3\%[/latex] interest and convert it to principal.
[latex]\color{blue}{\text{I/Y}\;}\color{black}{\text{is Nominal Interest Rate per Year:}}[/latex] A compound interest rate consists of two elements: a nominal number for the annual interest rate, known as the nominal interest rate, and words that state the compounding frequency. For example, a [latex]12\%[/latex] compounded quarterly interest rate is interpreted to mean that you accumulate [latex]12\%[/latex] nominal interest per year but the interest is converted to principal every compounding period, or every three months. The word nominal is used because if compounding occurs more than once per year, the true amount of interest that you earn per year is greater than the nominal interest rate.
[latex]\color{green}{\text{C/Y}\;}\color{black}{\text{is Compounds per Year:}}[/latex] The determination of [latex]\text{C/Y}[/latex] involves two key concepts: Compounding Period and Compounding Frequency. The Compounding Period is the amount of time that elapses between the dates of successive conversions of interest to principal is known as the compounding period (for example, a quarterly compounded interest rate converts interest to principal every three months, therefore, the compounding period is three months). The Compounding Frequency is the number of compounding periods in a complete year (for example, a quarterly compounded interest rate compounds every three months, or [latex]\text{C/Y}=4[/latex] times in a single year).
Concept Check
Example 6.1.1
Calculate the periodic interest rate, [latex]i[/latex], for the following nominal interest rates:
- [latex]9\%[/latex] compounded monthly
- [latex]6\%[/latex] compounded quarterly
Solution
Step 1: Given information:
- [latex]\text{I/Y}=9\%[/latex]; [latex]\text{C/Y}=\text{monthly}=12\;\text{times per year}[/latex]
- [latex]\text{I/Y}=6\%[/latex]; [latex]\text{C/Y}=\text{quarterly}=4\;\text{times per year}[/latex]
Step 2: For each question apply Formula 6.1.
a.
[latex]\begin{eqnarray*}I&=&\frac{\text{Nominal Rate (I/Y)}}{\text{Compounds per Year (C/Y)}}\\[1ex]I&=&\frac{9\%}{12}\\[1ex]I&=&0.75\%\;\text{per month}\end{eqnarray*}[/latex]
Nine percent compounded monthly is equal to a periodic interest rate of [latex]0.75\%[/latex] per month. This means that interest is converted to principal [latex]12[/latex] times throughout the year at the rate of [latex]0.75\%[/latex] each time.
b.
[latex]\begin{eqnarray*}I&=&\frac{\text{Nominal Rate (I/Y)}}{\text{Compounds per Year (C/Y)}}\\[1ex]I&=&\frac{6\%}{4}\\[1ex]I&=&1.5\%\; \text{per quarter}\end{eqnarray*}[/latex]
Step 3: Write as a statement.
Six percent compounded quarterly is equal to a periodic interest rate of [latex]1.5\%[/latex] per quarter. This means that interest is converted to principal [latex]4[/latex] times (every three months) throughout the year at the rate of [latex]1.5\%[/latex] each time.
Example 6.1.2
Calculate the nominal interest rate, [latex]\text{I/Y}[/latex], for the following periodic interest rates:
- [latex]0.58\overline{3}\%[/latex] per month
- [latex]0.05\%[/latex] per day
Solution
Step 1: Given information:
- [latex]i=0.58\overline{3}\%[/latex]; [latex]\text{C/Y}=\text{monthly}=12\;\text{times per year}[/latex]
- [latex]i=0.05\%[/latex]; [latex]\text{C/Y}=\text{daily}=365\;\text{times per year}[/latex]
Step 2: For each question, apply Formula 6.1 and rearrange for the nominal rate, [latex]\text{I/Y}[/latex].
a.
[latex]\begin{eqnarray*}\text{I/Y}&=&i\times \text{C/Y}\\\text{I/Y}&=&0.58\overline{3}\times12\\\text{I/Y}&=&7\%\end{eqnarray*}[/latex]
A periodic interest rate of [latex]0.58\overline{3}[/latex] per month is equal to a nominal interest rate of [latex]7\%[/latex] compounded monthly.
b.
[latex]\begin{eqnarray*}\text{I/Y}&=&i\times \text{C/Y}\\\text{I/Y}&=&0.05\times365\\\text{I/Y}&=&18.25\%\end{eqnarray*}[/latex]
Step 3: Write as a statement.
A periodic interest rate of [latex]0.05\%[/latex] per day is equal to a nominal interest rate of [latex]18.25\%[/latex] compounded daily.
Example 6.1.3
Calculate the compounding frequency ([latex]\text{C/Y}[/latex]) for the following nominal and periodic interest rates:
- [latex]\text{nominal interest rate}=6\%[/latex], [latex]\text{periodic interest rate}=3\%[/latex]
- [latex]\text{nominal interest rate}=9\%[/latex], [latex]\text{periodic interest rate}=2.25\%[/latex]
Solution
Step 1: Given information:
- [latex]\text{I/Y}=6\%[/latex]; [latex]i=3\%[/latex]
- [latex]\text{I/Y}=9\%[/latex]; [latex]i=2.25\%[/latex]
Step 2: For each question, apply Formula 6.1[latex]\begin{align*}i=\frac{\text{I/Y}}{\text{C/Y}}\end{align*}[/latex] and rearrange for the compounding frequency, [latex]\text{C/Y}[/latex].
a.
[latex]\begin{eqnarray*}\text{C/Y}&=&\frac{\text{I/Y}}{i}\\[1ex]\text{C/Y}&=&\frac{6\%}{3\%}\\[1ex]\text{C/Y}&=&2\;\text{compounds per year}\\\text{C/Y}&=&\text{semi-annually}\end{eqnarray*}[/latex]
For the nominal interest rate of [latex]6\%[/latex] to be equal to a periodic interest rate of [latex]3\%[/latex], the compounding frequency must be twice per year, which means a compounding period of every six months, or semi-annually.
b.
[latex]\begin{eqnarray*}\text{C/Y}&=&\frac{\text{I/Y}}{i}\\[1ex]\text{C/Y}&=&\frac{9\%}{2.25\%}\\[1ex]\text{C/Y}&=&4\;\text{compounds per year}\\\text{C/Y}&=&\text{quarterly}\end{eqnarray*}[/latex]
Step 3: Write as a statement.
For the nominal interest rate of [latex]9\%[/latex] to be equal to a periodic interest rate of [latex]2.25\%[/latex], the compounding frequency must be four times per year, which means a compounded period of every three months, or quarterly.
Section 6.1 Exercises
In each of the exercises that follow, try them on your own. Full solutions are available should you get stuck.
- Calculate the periodic interest rate if the nominal interest rate is [latex]7.75\%[/latex] compounded monthly.
Solution
[latex]\begin{eqnarray*}i&=&\frac{\text{I/Y}}{\text{C/Y}}\\[1ex]i&=&\frac{7.75\%}{12}\\[1ex]i&=&0.6458\%\;\text{per month}\end{eqnarray*}[/latex]
The periodic interest rate is [latex]0.65\%[/latex].
- Calculate the compounding frequency for a nominal interest rate of [latex]9.6\%[/latex] if the periodic interest rate is [latex]0.8\%[/latex].
Solution
[latex]\begin{eqnarray*}\text{C/Y}&=&\text{I/Y}\times i\\\text{C/Y}&=&9.6\%\times0.8\%\\\text{C/Y}&=&12\;\text{(monthly)}\end{eqnarray*}[/latex]
The compounding frequency is [latex]12[/latex] (monthly).
- Calculate the nominal interest rate if the periodic interest rate is [latex]2.0875\%[/latex] per quarter.
Solution
[latex]\begin{eqnarray*}\text{I/Y}&=&i\times\text{C/Y}\\\text{I/Y}&=&2.0875\%\times4\\\text{I/Y}&=&8.35\%\;\text{compounded quarterly}\end{eqnarray*}[/latex]
The nominal interest rate is [latex]8.35\%[/latex] compounded quarterly.
- After a period of three months, Alese saw one interest deposit of [latex]\$176.40[/latex] for a principal of [latex]\$9,800[/latex]. What nominal rate of interest is Alese earning?
Solution
Step 1: First convert the interest amount into a periodic interest rate per quarter.
[latex]\begin{eqnarray*}\text{Portion}&=&\text{Rate}\times\text{Base}\\I&=&i\times PV\\\$176.40&=&i\times\$9,800\\[1ex]i&=&\frac{\$176.40}{\$9,800}\\[1ex]i&=&0.018\;\text{or}\;1.8\%\;\text{per quarter}\end{eqnarray*}[/latex]
Step 2: Now convert the result in Step 1 to a nominal rate.
[latex]\begin{eqnarray*}\text{I/Y}&=&i\times\text{C/Y}\\\text{I/Y}&=&1.8\%\times4\\\text{I/Y}&=&7.2\%\;\text{compounded quarterly}\end{eqnarray*}[/latex]
Alese is earning [latex]7.2\%[/latex] compounded quarterly.
THE FOLLOWING LATEX CODE IS FOR FORMULA TOOLTIP ACCESSIBILITY. NEITHER THE CODE NOR THIS MESSAGE WILL DISPLAY IN BROWSER.[latex]\begin{align*}i=\frac{\text{I/Y}}{\text{C/Y}}\end{align*}[/latex]
Attribution
“9.1: Compound Interest and Fundamentals” from Business Math: A Step-by-Step Handbook Abridged by Sanja Krajisnik; Carol Leppinen; and Jelena Loncar-Vines is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.
